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Theorem wwlktovf1 13548
 Description: Lemma 2 for wrd2f1tovbij 13551. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
Hypotheses
Ref Expression
wrd2f1tovbij.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
wrd2f1tovbij.r 𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
wrd2f1tovbij.f 𝐹 = (𝑡𝐷 ↦ (𝑡‘1))
Assertion
Ref Expression
wwlktovf1 𝐹:𝐷1-1𝑅
Distinct variable groups:   𝑡,𝐷   𝑃,𝑛,𝑡,𝑤   𝑡,𝑅   𝑛,𝑉,𝑡,𝑤   𝑛,𝑋,𝑤
Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐹(𝑤,𝑡,𝑛)   𝑋(𝑡)

Proof of Theorem wwlktovf1
Dummy variables 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd2f1tovbij.d . . 3 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
2 wrd2f1tovbij.r . . 3 𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
3 wrd2f1tovbij.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡‘1))
41, 2, 3wwlktovf 13547 . 2 𝐹:𝐷𝑅
5 fveq1 6102 . . . . . 6 (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6 fvex 6113 . . . . . 6 (𝑥‘1) ∈ V
75, 3, 6fvmpt 6191 . . . . 5 (𝑥𝐷 → (𝐹𝑥) = (𝑥‘1))
8 fveq1 6102 . . . . . 6 (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9 fvex 6113 . . . . . 6 (𝑦‘1) ∈ V
108, 3, 9fvmpt 6191 . . . . 5 (𝑦𝐷 → (𝐹𝑦) = (𝑦‘1))
117, 10eqeqan12d 2626 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
1312eqeq1d 2612 . . . . . . 7 (𝑤 = 𝑥 → ((#‘𝑤) = 2 ↔ (#‘𝑥) = 2))
14 fveq1 6102 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
1514eqeq1d 2612 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
16 fveq1 6102 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
1714, 16preq12d 4220 . . . . . . . 8 (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
1817eleq1d 2672 . . . . . . 7 (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
1913, 15, 183anbi123d 1391 . . . . . 6 (𝑤 = 𝑥 → (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
2019, 1elrab2 3333 . . . . 5 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
21 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑦 → (#‘𝑤) = (#‘𝑦))
2221eqeq1d 2612 . . . . . . 7 (𝑤 = 𝑦 → ((#‘𝑤) = 2 ↔ (#‘𝑦) = 2))
23 fveq1 6102 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
2423eqeq1d 2612 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
25 fveq1 6102 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
2623, 25preq12d 4220 . . . . . . . 8 (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
2726eleq1d 2672 . . . . . . 7 (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
2822, 24, 273anbi123d 1391 . . . . . 6 (𝑤 = 𝑦 → (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
2928, 1elrab2 3333 . . . . 5 (𝑦𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
30 simp1 1054 . . . . . . . . . 10 (((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (#‘𝑥) = 2)
3130adantl 481 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (#‘𝑥) = 2)
32 simp1 1054 . . . . . . . . . . 11 (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → (#‘𝑦) = 2)
3332eqcomd 2616 . . . . . . . . . 10 (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → 2 = (#‘𝑦))
3433adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (#‘𝑦))
3531, 34sylan9eq 2664 . . . . . . . 8 (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (#‘𝑥) = (#‘𝑦))
3635adantr 480 . . . . . . 7 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (#‘𝑥) = (#‘𝑦))
37 simp2 1055 . . . . . . . . . . 11 (((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (𝑥‘0) = 𝑃)
3837adantl 481 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
39 simp2 1055 . . . . . . . . . . . 12 (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → (𝑦‘0) = 𝑃)
4039eqcomd 2616 . . . . . . . . . . 11 (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → 𝑃 = (𝑦‘0))
4140adantl 481 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
4238, 41sylan9eq 2664 . . . . . . . . 9 (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥‘0) = (𝑦‘0))
4342adantr 480 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘0) = (𝑦‘0))
44 simpr 476 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘1) = (𝑦‘1))
45 oveq2 6557 . . . . . . . . . . . . 13 ((#‘𝑥) = 2 → (0..^(#‘𝑥)) = (0..^2))
46 fzo0to2pr 12420 . . . . . . . . . . . . 13 (0..^2) = {0, 1}
4745, 46syl6eq 2660 . . . . . . . . . . . 12 ((#‘𝑥) = 2 → (0..^(#‘𝑥)) = {0, 1})
4847raleqdv 3121 . . . . . . . . . . 11 ((#‘𝑥) = 2 → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖)))
49 c0ex 9913 . . . . . . . . . . . 12 0 ∈ V
50 1ex 9914 . . . . . . . . . . . 12 1 ∈ V
51 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
52 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
5351, 52eqeq12d 2625 . . . . . . . . . . . 12 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
54 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑥𝑖) = (𝑥‘1))
55 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 1 → (𝑦𝑖) = (𝑦‘1))
5654, 55eqeq12d 2625 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
5749, 50, 53, 56ralpr 4185 . . . . . . . . . . 11 (∀𝑖 ∈ {0, 1} (𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
5848, 57syl6bb 275 . . . . . . . . . 10 ((#‘𝑥) = 2 → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
59583ad2ant1 1075 . . . . . . . . 9 (((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
6059ad3antlr 763 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
6143, 44, 60mpbir2and 959 . . . . . . 7 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → ∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖))
62 simpl 472 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → 𝑥 ∈ Word 𝑉)
63 simpl 472 . . . . . . . . . 10 ((𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑦 ∈ Word 𝑉)
6462, 63anim12i 588 . . . . . . . . 9 (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
6564adantr 480 . . . . . . . 8 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
66 eqwrd 13201 . . . . . . . 8 ((𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
6765, 66syl 17 . . . . . . 7 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ∀𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) = (𝑦𝑖))))
6836, 61, 67mpbir2and 959 . . . . . 6 ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → 𝑥 = 𝑦)
6968ex 449 . . . . 5 (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
7020, 29, 69syl2anb 495 . . . 4 ((𝑥𝐷𝑦𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
7111, 70sylbid 229 . . 3 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7271rgen2a 2960 . 2 𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
73 dff13 6416 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
744, 72, 73mpbir2an 957 1 𝐹:𝐷1-1𝑅
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  {cpr 4127   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  ..^cfzo 12334  #chash 12979  Word cword 13146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154 This theorem is referenced by:  wwlktovf1o  13550
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